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Given a class of structured matrices $\mathbb{S}$, we identify pairs of vectors $x,b$ for which there exists a matrix $A\in\mathbb{S}$ such that $Ax=b$, and we also characterize the set of all matrices $A\in\mathbb{S}$ mapping $x$ to $b$. The structured classes we consider are the Lie and Jordan algebras associated with orthosymmetric scalar products. These include (skew-)symmetric, (skew-)Hamiltonian, pseudo(skew-)Hermitian, persymmetric, and perskew-symmetric matrices. Structured mappings with extremal properties are also investigated. In particular, structured mappings of minimal rank are identified and shown to be unique when rank one is achieved. The structured mapping of minimal Frobenius norm is always unique, and explicit formulas for it and its norm are obtained. Finally the set of all structured mappings of minimal 2-norm is characterized. Our results generalize and unify existing work, answer a number of open questions, and provide useful tools for structured backward error investigations.

Structured Mapping Problems for Matrices Associated with Scalar Products. Part I: Lie and Jordan Algebras